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HPD) Is One of the Most Exciting Recent Break- Throughs in Homological Algebra and Algebraic Geometry

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HPD) Is One of the Most Exciting Recent Break- Throughs in Homological Algebra and Algebraic Geometry HOMOLOGICAL PROJECTIVE DUALITY FOR DETERMINANTAL VARIETIES MARCELLO BERNARDARA, MICHELE BOLOGNESI, AND DANIELE FAENZI Abstract. In this paper we prove Homological Projective Duality for cate- gorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m × n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective vari- ety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties. 1. Introduction Homological Projective Duality (HPD) is one of the most exciting recent break- throughs in homological algebra and algebraic geometry. It was introduced by A. Kuznetsov in [26] and its goal is to generalize classical projective duality to a ho- mological framework. One of the important features of HPD is that it offers a very important tool to study the bounded derived category of a projective variety to- gether with its linear sections, providing interesting semiorthogonal decompositions as well as derived equivalences, cf. [22, 23, 28, 3, 27]. Roughly speaking, two (smooth) varieties X and Y are HP-dual if X has an am- ple line bundle OX (1) giving a map X ! PW , Y has an ample line bundle OY (1) giving a map Y ! PW _, and X and Y have dual semiorthogonal decompositions (called Lefschetz decompositions) compatible with the projective embedding. In this case, given a generic linear subspace L ⊂ W and its orthogonal L? ⊂ W _, one can consider the linear sections XL and YL of X and Y respectively. Kuznetsov b shows the existence of a category CL which is admissible both in D (XL) and in b D (YL), and whose orthogonal complement is given by some of the components of the Lefschetz decompositions of Db(X) and Db(Y ) respectively. That is, both b b D (XL) and D (YL) admit a semiorthogonal decomposition by a \Lefschetz" com- ponent, obtained via iterated hyperplane sections, and a common \nontrivial" or \primitive" component. HPD is closely related to classical projective duality: [26, Theorem 7.9] states that the critical locus of the map Y ! PW _ coincides with the classical projective dual of X. The main technical issue of this fact is that one has to take into account singular varieties, since the projective dual of a smooth variety is seldom smooth - e.g. the dual of certain Grassmannians are singular Pfaffian varieties [12]. On the other hand, derived (dg-enhanced) categories should provide a so-called categorical or non-commutative resolution of singularities ([30, 37]). Roughly speaking, one D. F. partially supported by GEOLMI ANR-11-BS03-0011. 1 2 MARCELLO BERNARDARA, MICHELE BOLOGNESI, AND DANIELE FAENZI b needs to find a sheaf of OY -(dg)-algebras R such that the category D (Y; R) of bounded complexes of coherent R-modules is proper, smooth and R is locally Morita-equivalent to some matrix algebra over OY (this latter condition translates the fact that the resolution is birational). In the case where Y is singular, one of the most difficult tasks in proving HPD is to provide such a resolution with the required Lefschetz decomposition (for example, see [27, x4.7]). On the other hand, given a non-smooth variety, it is a very interesting question to provide such resolutions and study their properties such as crepancy, minimality and so forth. The main application of HPD is that it is a direct method to produce semiorthog- onal decompositions for projective varieties with non-trivial canonical sheaf, and derived equivalences for Calabi-Yau varieties. The importance of this application is due to the fact that determining whether a given variety admits or not a semiorthog- onal decomposition is a very hard problem in general. Notice that there are cases where it is known that the answer to this question is negative, for example if X has trivial canonical bundle [13, Ex. 3.2], or if X is a curve of positive genus [34]. On the other hand, if X is Fano, then any line bundle is exceptional and gives then a semiorthogonal decomposition. Almost all the known cases of semiorthogonal de- compositions of Fano varieties described in the literature (see, e.g., [22, 29, 23, 6, 3]) can be obtained via HPD or its relative version. Derived equivalences of Calabi-Yau (CY for short) varieties have deep geometri- cal insight. First of all, it was shown by Bridgeland that birational CY-threefolds are derived equivalent [14]. The converse in not true: the first example - that has been shown to be also a consequence of HPD in [25] - was displayed by Borisov and Caldararu in [12]. Besides their geometric relevance, derived equivalences between CY varieties play an important role in theoretical physics. First of all, Kontsevich's homological mirror symmetry conjectures an equivalence between the bounded derived category of a CY-threefold X and the Fukaya category of its mirror. More recently, it has been conjectured that homological projective duality should be realized physically as phases of abelian gauged linear sigma models (GLSM) (see [17] and [2]). As an example, denote by X and Y the pair of equivalent CY{threefolds consid- ered by Borisov and Caldararu. Rødland [36] argued that the families of X's and Y 's (letting the linear section move in the ambient space) seem to have the same mirror variety Z (a more string theoretical argument has been given recently by Hori and Tong [18]). The equivalence between X and Y would then fit Kontsevich's Homological Mirror Symmetry conjecture via the Fukaya category of Z. It is thus fair to say that HPD plays an important role in understanding these questions and potentially providing new examples. Notice in particular that some determinantal cases were considered in [20]. In this paper, we describe new families of HP Dual varieties. We consider two vector spaces U and V of dimension m and n respectively with m ≤ n. Let G = G(U; r) denote the Grassmannian of r-dimensional quotients of U, set Q and U for the universal quotient and sub-bundle respectively. Let X := P(V ⊗ Q) and Y := P(V _ ⊗ U _), for any 0 < r < m. Let p : X ! G and q : Y ! G be the natural projections. Set HX and HY for the relatively ample tautological divisors HPD FOR DETERMINANTAL VARIETIES 3 on X and Y . Orlov's result [35] provides semiorthogonal decompositions b ∗ b ∗ b D (X ) = hp D (G); : : : ; p D (G) ⊗ OX ((rn − 1)HX )i; (1.1) b ∗ b ∗ b D (Y ) = hq D (G) ⊗ OY (((r − m)n + 1)HY ); : : : ; q D (G)i: Theorem 3.5. In the previous notation, X and Y with Lefschetz decompositions (1.1) are HP-dual. The proof of the previous result is a consequence of Kuznetsov's HPD for pro- jective bundles generated by global sections (see [26, x8]). Here, the spaces of global sections of OX (HX ) and OY (HY ) sheaves are, respectively, W = V ⊗U and W _ = V _ ⊗ U _. The main interest of Theorem 3.5 is that X is known to be the resolution of the variety Z r of m × n matrices of rank at most r. Write such a matrix as M : U ! V _. Then Z r is naturally a subvariety of PW , which is singular in general, with resolution f : X ! Z r. Dually, g : Y ! Z m−r is a desingularization of the variety of m×n matrices of corank at least r. Theorem 3.5 provides the categorical framework to describe HPD between the classical projectively dual varieties Z r and Z m−r (see, e.g., [39]). In the affine case, categorical resolutions for determinantal varieties have been constructed by Buchweitz, Leuschke and van den Bergh [15, 16]. Such resolution is crepant if m = n (that is, in the case where Z r has Gorenstein singularities). The starting point is Kapranov's construction of a full strong exceptional collection on Grassmannians [21]. One can use the decompositions in exceptional objects (1.1) to produce a sheaf of algebras R0 and a categorical resolution of singularities Db(Z r; R0) ' Db(X ). For simplicity, we will denote by R0 the algebra on any of the determinantal varieties Z r (forgetting about the dependence of R0 on the rank r). This gives a geometrically deeper version of Theorem 3.5. Theorem 3.6. In the previous notations, Z r admits a categorical resolution of singularities Db(Z r; R0), which is crepant if m = n. Moreover, Db(Z r; R0) and Db(Z m−r; R0) are HP-dual. Once the equivalence Db(Z r; R0) ' Db(X ) constructed, Theorem 3.6 is proved by applying directly Theorem 3.5. However, the geometric relevance of Theorem 3.6, and its difference with Theorem 3.5, is that it shows HPD directly on noncom- mutative structures over the determinantal varieties Z r and Z m−r with respect to their natural embedding in PW and PW _ respectively. That is, these natural smooth and proper noncommutative scheme structures are well-behaved with re- spect to projective duality and hyperplane sections. Finally, notice that whenever r L m−r we pick a smooth linear section ZL of Z (or a smooth section Z of Z ), the 0 restriction to ZL of the sheaf R is Morita-equivalent to OZL , so that we get the b derived category D (ZL) of the section itself.
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